My daughter has a 2001 Honda Accord EX with 126k on it. She bought it 7 years ago with only 17K on it(was owned by an elderly lady who only drove it to church and to the stores). The engine runs great, I think the engine will outlast the body because the body is beginning to show its age more quickly now. For the past year she has been dealing with it starting when it wants to. There are times she has no problems with it for a month, but then all of a sudden, it will only crank but not turn over. She has to sit there for several minutes up to a 1/2 hour waiting and then tries it again and it will usually start. About 6 months ago it wouldn't do anything so we pulled it to the shop and he reprogrammed her fob cause he couldn't find anything wrong with it and it worked just fine after that. But for the past 3 weeks, it started acting up again with the cranking but not turning over. So she switched keys and was using her spare key/fob and it was starting fine. But last friday, it wouldn't start. Just kept cranking and not turning over. And my husband tried as well and nothing. So we let it sit til Sunday and were going to pull it to another shop and drop it off, but when I got in it, it started right up. So I drove it to the shop. Its just a small country shop with a mechanic there. Every time we've brought it to him, its started every time for him so there was nothing he could do with out it acting up. It drove just fine there. I even tried to start it again once we arrived and it started fine. Any ideas on what this could be? She has a 9 month old baby so I don't need them stranded with hot texas summer coming. She's trying to finish college next month and hopefully find a full time job after then she'll be able to buy a newer car for herself, but she needs this one to work until then.

 

No check Engine Light?
Anyway, for what it's worth, on my (now my son's) '99 LX V6, the fuel pump relay was replaced, curing a symptom similar to your description. Long ago, but the honda parter number should be 39400-S84-003.
Technically, there should be a way to test that part, but it may take as much expense to diagnose as it will to just replace it.